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arXiv:1612.09464

On Three Imaginary-time Path Integral Formulas with Magnetic Fields in Relativistic Quantum Mechanics

Takashi Ichinose

Published Dec 30, 2016• 0 citations
arXivHTMLPDF

Abstract

Three magnetic relativistic Schrödinger operators are considered, corresponding to the classical relativistic Hamiltonian symbol with both magnetic vector and electric scalar potentials. Path integral representations for the solutions of their respective imaginary-time relativistic Schrödinger equations, i.e. heat equations are given in two ways. The one is by means of the probability path space measure coming from the Lévy process concerned, and the other is through time-sliced approximation with Chernoff's theorem.

DOI
10.48550/arxiv.1612.09464

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Three magnetic relativistic Schrödinger operators are considered, corresponding to the classical relativistic Hamiltonian symbol with both magnetic vector and electric scalar potentials. Path integral representations for the solutions of their respective imaginary-time relativistic Schrödinger equations, i.e. heat equations are given in two ways. The one is by means of the probability path space measure coming from the Lévy process concerned, and the other is through time-sliced approximation with Chernoff's theorem.
This paper is a review of path integral representations for semigroups {exp(-t H_r/ħ)}{t≥0} where H_r's are relativistic quantum Hamiltonians. We consider three different cases: in the first one Hr is a relativistic Schrödinger operator, in the second is the Hamiltonian associated to Klein-Gordon equation and in the third is that coming from the Dirac one. The paths are trajectories of diffusion processes and, for the Dirac case, they differ from previous constructions based on jump Markov processes. All formulas describe one relativistic particle in an external static electromagnetic field therefore they are relativistic versions of the Feymann-Kac representation for Schrödinger semigroups.
The main aim of the present paper is using a Chernoff theorem (i.e., the Chernoff formula) to formulate and to prove some rigorous results on representations for solutions of Schrödinger equations by the Hamiltonian Feynman path integrals (=Feynman integrals over trajectories in the phase space). The corresponding theorem is related to the original (Feynman) approach to Feynman path integrals over trajectories in the phase space in much the same way as the famous theorem of Nelson is related to the Feynman approach to the Feynman path integral over trajectories in the configuration space. We also give a representation for solutions of some Schrödinger equations by a series which represents an integral with respect to the complex Poisson measure on trajectories in the phase space.
The ordinary Feynman-Kac-Itô formula gives a path integral representation of the Schrödinger semi-groups. We discuss here an analogous probabilistic expression for the (positive energy) solutions of the imaginary-time Klein-Gordon equation in a static external electromagnetic field. When the external field is not purely magnetic, our result is different from the path integral associated to the semi-group t ∊ [0, + ∞ ) → exp[− t ( H − mc 2 )/ℏ] of a relativistic Schrödinger operator H = { c 2 [− i ℏ∇−( e/c ) A ] 2 + m 2 c 4 } 1/2 + V .
Solutions to the Schrödinger, heat and stochastic Schrödinger equation with rather general potentials are represented, both in x- and p-representations, as integrals over the path space with respect to σ-finite measures. In the case of x-representation, the corresponding measure is concentrated on the Cameron–Martin Hilbert space of curves with L2-integrable derivatives. The case of the Schrödinger equation is treated by means of a regularization based on the introduction of either complex times or continuous non-demolition observations.
Evolution semigroups generated by pseudo-differential operators are considered. These operators are obtained by different (parameterized by a number τ) procedures of quantization from a certain class of functions (or symbols) defined on the phase space. This class contains Hamilton functions of particles with variable mass in magnetic and potential fields and more general symbols given by the Lévy-Khintchine formula. The considered semigroups are represented as limits of n-fold iterated integrals when n tends to infinity. Such representations are called Feynman formulae. Some of these representations are constructed with the help of another pseudo-differential operator, obtained by the same procedure of quantization; such representations are called Hamiltonian Feynman formulae. Some representations are based on integral operators with elementary kernels; these are called Lagrangian Feynman formulae. Langrangian Feynman formulae provide approximations of evolution semigroups, suitable for direct computations and numerical modeling of the corresponding dynamics. Hamiltonian Feynman formulae allow to represent the considered semigroups by means of Feynman path integrals. In the article, a family of phase space Feynman pseudomeasures corresponding to different procedures of quantization is introduced. The considered evolution semigroups are represented as phase space Feynman path integrals with respect to these Feynman pseudomeasures, i.e., different quantizations correspond to Feynman path integrals with the same integrand but with respect to different pseudomeasures. This answers Berezin’s problem of distinguishing a procedure of quantization on the language of Feynman path integrals. Moreover, the obtained Lagrangian Feynman formulae allow also to calculate these phase space Feynman path integrals and to connect them with some functional integrals with respect to probability measures.